To determine which trinomial is factored among the given options, we need to examine each trinomial and its factors.
Let's analyze each trinomial in detail.
1. For [tex]\( x^2 + 3x - 6 \)[/tex]:
We can attempt to factorize it as follows:
- The trinomial [tex]\( x^2 + 3x - 6 \)[/tex] does not factor neatly into integer factors. Hence, it remains:
[tex]\[ x^2 + 3x - 6 \][/tex]
2. For [tex]\( x^2 + 5x - 6 \)[/tex]:
We can factorize it as follows:
- Look for two numbers that multiply to [tex]\(-6\)[/tex] and add to [tex]\(5\)[/tex]. These numbers are [tex]\(6\)[/tex] and [tex]\(-1\)[/tex].
[tex]\[ x^2 + 5x - 6 = (x + 6)(x - 1) \][/tex]
3. For [tex]\( x^2 + 3x - 2 \)[/tex]:
We can attempt to factorize it as follows:
- The trinomial [tex]\( x^2 + 3x - 2 \)[/tex] does not factor neatly into integer factors. Hence, it remains:
[tex]\[ x^2 + 3x - 2 \][/tex]
4. For [tex]\( x^2 + x - 6 \)[/tex]:
We can factorize it as follows:
- Look for two numbers that multiply to [tex]\(-6\)[/tex] and add to [tex]\(1\)[/tex]. These numbers are [tex]\(3\)[/tex] and [tex]\(-2\)[/tex].
[tex]\[ x^2 + x - 6 = (x + 3)(x - 2) \][/tex]
Comparing these factorizations with the given results [tex]\([x^2 + 3x - 6, (x - 1)(x + 6), x^2 + 3x - 2, (x - 2)(x + 3)]\)[/tex], we recognize the patterns correspond to our factorization steps.
Hence, the correct trinomial that has been factored is [tex]\( x^2 + 5x - 6 \)[/tex], and it factors to:
[tex]\[ (x - 1)(x + 6) \][/tex]
